Editing Matrix decomposition (section)

=== Scale-invariant decompositions ===

Refers to variants of existing matrix decompositions, such as the SVD, that are invariant with respect to diagonal scaling.

*Applicable to: ''m''-by-''n'' matrix ''A''.
*Unit-Scale-Invariant Singular-Value Decomposition: <math>A=DUSV^*E</math>, where ''S'' is a unique nonnegative [[diagonal matrix]] of scale-invariant singular values,  ''U'' and ''V'' are [[unitary matrix|unitary matrices]], <math>V^*</math> is the [[conjugate transpose]] of ''V'', and positive diagonal matrices ''D'' and ''E''.
*Comment: Is analogous to the SVD except that the diagonal elements of ''S'' are invariant with respect to left and/or right multiplication of ''A'' by arbitrary nonsingular diagonal matrices, as opposed to the standard SVD for which the singular values are invariant with respect to left and/or right multiplication of ''A'' by arbitrary unitary matrices.
*Comment: Is an alternative to the standard SVD when invariance is required with respect to diagonal rather than unitary transformations of ''A''.
*Uniqueness: The scale-invariant singular values of <math>A</math> (given by the diagonal elements of ''S'') are always uniquely determined. Diagonal matrices ''D'' and ''E'', and unitary ''U'' and ''V'', are not necessarily unique in general.
*Comment: ''U'' and ''V'' matrices are not the same as those from the SVD.

Analogous scale-invariant decompositions can be derived from other matrix decompositions; for example, to obtain scale-invariant eigenvalues.<ref>{{citation|last=Uhlmann |first=J.K. |title=A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations |journal=SIAM Journal on Matrix Analysis and Applications |year=2018 |volume=239 |issue=2 |pages=781–800 |doi=10.1137/17M113890X }}</ref><ref>{{citation|last=Uhlmann |first=J.K. |title=A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity |journal=IEEE Control Systems Letters |issn=2475-1456 |year=2018 |volume=3 |pages=91–95 |doi=10.1109/LCSYS.2018.2854240 |arxiv=1804.07334 |s2cid=5031440 }}</ref>